The Fiftieth Anniversary of Chaos

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The Fiftieth Anniversary of Chaos

Several decades ago, Edward Lorenz noticed something seemingly minor: his computational weather models yielded very different results if he changed the input variables only a tiny bit. Lorenz published a paper on this topic in 1963 and that led to the modern field of chaos.

A chaotic attractor is the example par excellence of a chaotic set. A chaotic set has uncountably many chaotic trajectories; on such a set, any point that lies in the neighborhood of a given point will also, with probability one, give rise to a chaotic trajectory. Yet no matter the proximity of those two points, in the region between them will lie points of infinitely many periodic orbits. In mathematical parlance, the periodic orbits constitute a countable, zero-measure, but dense set of points embedded in the chaotic set, analogous to the rational numbers embedded in the set of real numbers. Not only will trajectories that lie on the attractor behave chaotically, any point lying within the attractor’s basin of attraction will also give rise to chaotic trajectories that converge to the attractor.

Of course, chaos is found in many places far beyond weather models:

As Motter and Campbell conclude, chaos is far more than something of interest only to mathematicians or physicists:

Chaos sets itself apart from other great revolutions in the physical sciences. In contrast to, say, relativity or quantum mechanics, chaos is not a theory of any particular physical phenomenon. Rather, it is a paradigm shift of all science, which provides a collection of concepts and methods to analyze a novel behavior that can arise in a wide range of disciplines.